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Probability Relations and Induction

This contribution addresses probability relations and hopes to establish a rational component to induction and partial beliefs.

In deductive logic, a conclusion is true just in case the premises are true. An example of this is a categorical syllogism: if all As are Bs, and all Bs are Cs, then it can be deduced that all As are Cs. In deductive logic the movement of reasoning is from general to specific.

With inductive reasoning, empirical observations are made and from these generalizations are induced. For example, if a person throws a brick at a person's head at 10 second intervals, eventually the person may get into the rhythm of ducking at the interval to avoid being hit by the brick. Maybe this goes on for 19 bricks. Maybe at the 20th interval the person ducks in anticipation but is surprised that no brick is thrown. Point: it does not follow that because the brick was thrown 19 times that it will be thrown on the 20th. Similarly, a person can have an intuition that the world is flat, and from this make the claim that the world is flat. Yet we know that this would be a fallacious belief as scientific observations, satellite imaging, and thought have given us much more evidence to think the earth is a sphere. Yet, inductive arguments can be better or worse depending on the evidence in its favor. Indeed, based on a series of precedents one can have a highly educated guess that the brick will be thrown at the 20th interval, that there will not be a tornado in Toronto tomorrow, and that the Sun will rise tomorrow, but these are not and are not supposed to be deductively valid. As a result, the logical thinker can never be certain that the sun will rise tomorrow the way he can be certain that a categorical or disjunctive syllogism is logically valid.

Yet, to say that inductive arguments are non-rational would be misleading. In order to give a rational component to induction we will need to bring in probability. In short, we will need to extend our notion of a logical relation to include probability relations in order to establish how intuitive knowledge can form the basis of rational beliefs that fall short of knowledge. As Keynes has pointed out, a probability statement expresses a logical relationship between proposition ‘p’ and proposition ‘h’ (h is usually a conjunction of propositions). A man who knows ‘h’ and perceives a logical relationship between p and h is justified in believing p with a degree of belief which corresponds to that of the logical relationship. As such, probability is a logical relation holding between propositions which are similar to, although weaker than, logical consequences.

A proposition may be probable to a certain set of data and highly improbable to another set of data which includes the first set as a part. For example, if the only fact you know about a boxer is that his fight ended 30 seconds into the first round, it is quite probable with respect to these data that someone got knocked out or technically knocked out. If you find out after there was a power outage at ringside during his fight, then the probability that someone got knocked out, on combined data, is smaller. We can also think through the following example: given that X is an inhabitant of Europe it follows that he lives either in Britain or in France or in Germany or…Here, one can assign probabilities to person X living in each state based on certain conditions—perhaps relative populations being one of those conditions. Indeed, conditions are critical. To every event there is a finite set of conditions relative to which the event is certain to happen or certain not to happen. If the evidence reveals more of these conditions, we shall say that we have a stronger case, and we can extract probabilities and assign relative values to consequences. Some have argued that one way of quantifying probability relations in a real sense, could be to have the degree of belief in a proposition 'p', which a particular man has at a particular time, by the rate at which he would be prepared to bet upon 'p' being true. In this sense, there is a continuum: falsehood<--l--l--l--l--> truth with various probabilities in between. It is argued, therefore, that the test of relative strength of belief can be quantified and thus measured by a bet a person is willing to stake on their belief.

All in on this point: in matters of induction we cannot hope for anything more than probable conclusions and, therefore, the logical principles of induction must be the laws of probability. In this way, we are able to link intuitions and partial beliefs to a rational base grounded in probabilities.

In deductive logic, a conclusion is true just in case the premises are true.

The conclusion of a deductive argument may be true even when the premises are false. For example,

Every hamster is a member of TypologyCentral
Provoker is a hamster
Therefore,
Provoker is a member of TypologyCentral

The deductive relation between premises and conclusion is merely that if the premises are true, then the conclusion is also true.

In deductive logic the movement of reasoning is from general to specific.

A deduction can be from a generalisation to a specific, from a generalisation to another generalisation, or a specific to another specific. In any case, this characterisation -- an attempt to make deduction appear the opposition of induction -- is rather misleading outside of basic syllogisms.

With inductive reasoning, empirical observations are made and from these generalizations are induced.

Unless we have stopped talking about logic and have started talking about psychology, inductions are never from observations. Premises are propositions, and while we may use them to describe some experience we had, there is no logical connection.

Yet we know that this would be a fallicious belief as scientific observations, satellite imaging, and thought have given us much more evidence to think the earth is a sphere.

A fallacy is an error of reasoning, not fact. A belief, by itself, cannot be fallicious, because a belief is not a kind of reasoning, but merely a claim, assertion, or proposition.

Yet, inductive arguments can be better or worse depending on the evidence in its favor. Indeed, based on a series of precedents one can have a highly educated guess that the brick will be thrown at the 20th interval, that there will not be a tornado in Toronto tomorrow, and that the Sun will rise tomorrow, but these are not and are not supposed to be deductively valid. As a result, the logical thinker can never be certain that the sun will rise tomorrow the way he can be certain that a categorical or disjunctive syllogism is logically valid.

Are you the arbiter of what people "can" and "can't" be certain about?

In any case, if it's just educated guessing, then why do you need induction at all? Why not just educatedly guess that the sun will rise tomorrow in the premises? Or more on point, what does putting such beliefs in a logical argument achieve in the first place?

Yet, to say that inductive arguments are non-rational would be misleading. In order to give a rational component to induction we will need to bring in probability. In short, we will need to extend our notion of a logical relation to include probability relations in order to establish how intuitive knowledge can form the basis of rational beliefs that fall short of knowledge. As Keynes has pointed out, a probability statement expresses a logical relationship between proposition ‘p’ and proposition ‘h’ (h is usually a conjunction of propositions). A man who knows ‘h’ and perceives a logical relationship between p and h is justified in believing p with a degree of belief which corresponds to that of the logical relationship. As such, probability is a logical relation holding between propositions which are similar to, although weaker than, logical consequences.

But evidence can provide no probabilistic support for a hypothesis beyond that part of the hypothesis which is equal to the evidence. In other words, attempting to attribute the increased probability to all parts of the hypothesis equally commits the fallacy of composition.

All in on this point: in matters of induction we cannot hope for anything more than probable conclusions and, therefore, the logical principles of induction must be the laws of probability. In this way, we are able to link intuitions and partial beliefs to a rational base grounded in probabilities.

The statement "there are at most a, b, and c ravens" is equivalent to "all ravens are not d, e, f, ...". In other words, unless the number of something is limited by pure logic, then positing such a limit in order to enable induction assumes a universal, and thus the underlying problem isn't solved.

A criticism that can be brought against everything ought not to be brought against anything.

The conclusion of a deductive argument may be true even when the premises are false. For example,

Every hamster is a member of TypologyCentral
Provoker is a hamster
Therefore,
Provoker is a member of TypologyCentral

.

What you want to say is that the syllogism is logically valid, which doesn't necessitate the conclusion's truth. Here, if one falsifies the premises, then the conclusion is not true because of its relation to the premises. In this particular case, the conclusion happens to be true for reasons outside of the syllogism itself.

Originally Posted by reason

The deductive relation between premises and conclusion is merely that if the premises are true, then the conclusion is also true.

Here, you've managed to conclude this part with basically restating what I said in other words: (Provoker "In deductive logic, a conclusion is true just in case the premises are true").

Originally Posted by reason

A deduction can be from a generalisation to a specific, from a generalisation to another generalisation, or a specific to another specific. In any case, this characterisation -- an attempt to make deduction appear the opposition of induction -- is rather misleading outside of basic syllogisms.

Aha! I think if we look at the bigger picture afforded by history deductive reasoning, if it is to yield fruit, tends to move from general to specific. For example, Newton induced his theory of gravity.Then in the 19th century other thinkers took Newton's theory (general principle) to deduce the existence, mass, position, and orbit of Neptune (specific conclusions) from perturbations in the observed orbit of Uranus (specific data). So, if your point is to be relevant, you would have to show that the movement from generalization to generalization and spefic to specific is not merely contained in a broader movement from general to specific. So long as these microdeductions are contained in a broader movement from general to specific, my statement remains true.

Originally Posted by reason

inductions are never from observations.

You don't really mean that do you? When a child induces the general claim that the world is flat because it appears that way, where do you think this comes from if not an observation?

Originally Posted by reason

A belief, by itself, cannot be fallicious, because a belief is not a kind of reasoning, but merely a claim, assertion, or proposition.

Ok this is a rather moot point, and you've isolated one point rather than considering the context in which it was used. By saying 'fallacious belief', it is taken to mean a belief that that is arrived at fallaciously--therefore a misleading belief. I suppose I should have explained myself more thoroughly in that the inductive reasoning is the part that is fallacious and that the belief that came out of that reasoning is false. In leaving this out, my assumption was that an intelligent audience would make these connections automatically.

Originally Posted by reason

In any case, if it's just educated guessing, then why do you need induction at all?.

Induction, experimentation, and a more empirical approach has historically been the source of many discoveries. The trouble with deductive reasoning is it is very limiting in that the conclusions are never beyond what is already contained in the premises. If inductive reasoning was never used, the human species would miss out on a lot of innovations, discoveries, and so forth. Orson Welles would have been disabled from experimenting during Citizen Kane, which revolutionized cinema. Newton would have been disabled from discovering gravity. In one sentence, if induction was abandoned many great discoveries would be foregone. Therefore given a weighty opportunity cost, perhaps you can detail why this would be a trade-off worth making?

Here, you've managed to conclude this part with basically restating what I said in other words: (Provoker "In deductive logic, a conclusion is true just in case the premises are true").

He's saying the conclusion is definitely true if the premises are true, but can also be true if the premises are false.

Aha! I think if we look at the bigger picture afforded by history deductive reasoning, if it is to yield fruit, tends to move from general to specific. For example, Newton induced his theory of gravity.Then in the 19th century other thinkers took Newton's theory (general principle) to deduce the existence, mass, position, and orbit of Neptune (specific conclusions) from perturbations in the observed orbit of Uranus (specific data). So, if your point is to be relevant, you would have to show that the movement from generalization to generalization and spefic to specific is not merely contained in a broader movement from general to specific. So long as these microdeductions are contained in a broader movement from general to specific, my statement remains true.

Here's a deductive argument.

A
B
therefore
A
B

Your conclusion is always going to be a subset of your premises (as long as you rewrite them accordingly), but it can also be a restatement of all of them.

So it can go from general to general, specific to specific, or general to specific, but never specific to general (that's induction).

Induction, experimentation, and a more empirical approach has historically been the source of many discoveries. The trouble with deductive reasoning is it is very limiting in that the conclusions are never beyond what is already contained in the premises. If inductive reasoning was never used, the human species would miss out on a lot of innovations, discoveries, and so forth. Orson Welles would have been disabled from experimenting during Citizen Kane, which revolutionized cinema. Newton would have been disabled from discovering gravity. In one sentence, if induction was abandoned many great discoveries would be foregone. Therefore given a weighty opportunity cost, perhaps you can detail why this would be a trade-off worth making?

Here's my question...I see your whole thing about probabilities changing based on new inputs. But what if your inputs equally support two theories? You've provided no explanation for how you would choose between them.

It's likely you've heard of the "grue" example, but I'll go over it anyway. Suppose the term "grue" means green before 2012 and blue after 2012. So if you look at a leaf, you could call that green or grue, and be equally correct. No matter how many things you've seen in the world, there's no way to distinguish between those two labels. So if you make the conclusion, "in february of 2012, leaves will be blue", that's just as likely based on the data you've seen as "in february of 2012, leaves will be green".

This is meant to illustrate the point that there are an infinite number of hypotheses that are consistent with a set of data. So not only do we need a way to modify the strength of our beliefs based on data we see, we also need to be able to explain why we prefer certain kinds of hypotheses when an infinite number are all consistent with the data we've gotten.

First, it is possible to validly infer a universal statement from a set of premises including only singular statements. For example,

a is y |= All x are y or not-y

This only works when the conclusion is a tautology (and, therefore, even follows from the empty set of premises), but should be noted as a caveat to previous comments.

Second, here is a simple explanation of why inductive logic does not exist. Take this simple induction:

a is y, b is y, c is y, d is y |- every x is y

Then define a new predicate z:

if x equals a, b, c, or d, and is y, then x is z; otherwise, if x is not-y, then x is z.

Therefore, "every x is z" is true if:

a is y, b is y, c is y, d is y, e is not-y, f is not-y, g is not-y, ad infinitum.

Since a, b, c, and d, are also z if y, then:

a is y, b is y, c is y, d is y |= a is z, b is z, c is z, d is z

Therefore,

a is y, b is y, c is y, d is y |- every x is z

However, "every x is y" and "every x is z" contradict each other. Therefore, two mutually inconsistent universals can be induced from the same set of premises. Using this procedure, we can define infinitely many predicates like z, all of which lead to mutually incompatible inductions. The only limit on what can be "induced" comes from deductive truth relations.

This argument is a pure logical analogue of the problem of theory-ladenness; it demonstrates the arbitrariness of inductive inference, and, therefore, the absence of an inductive logic -- which, by definition, is non-arbitrary. In other words, there is no way to define "inductive validity."

NOTE TO EVAN: The above is a formalised -- and stronger version -- of the grue problem of induction, but one which does not muddy the waters with issues of time, gems, and colours. Since it is purely formal, all it demonstrates is that induction is no logic -- it doesn't mean that people do not make generalisations.

A criticism that can be brought against everything ought not to be brought against anything.

"Every hamster is a member of TypologyCentral
Provoker is a hamster
Therefore,
Provoker is a member of TypologyCentral"

This argument is guilty of the informal logical fallacy of equivocation. Logic has two components, extensions and intensions. Extensions are concerned with how entities of discourse relate to each other. (In accordance to laws of reasoning, for instance, demorgan's law, modus ponens, modus tollens and so on.) Intensions are concerned with how the terms of discourse are defined.

When I say that provoker is a member of typologycentral, I define provoker as a person who made several posts on this forum. If I say that provoker is a hamster, I mean something entirely different. Hence, in the argument above, the conclusion is false. A hamster named provoker is not a member of typologycentral.

For this reason, the argument cited above is not an example of an argument with false conclusions leading to true premises and therefore does not prove that it is possible for a deductively valid argument to have false premises and a true conclusion. What we have here is an argument with false premises leading to a false conclusion. However, because the reasoning in question is deductively valid, the conclusion would have been true if the premises were also.

However, an argument with contradictory premises may be deductively valid and at the same time entail a false conclusion. Since anything follows from a contradiction, it is possible to derive a true conclusion from an argument with contradictory premises. If you construct the truth table for the following argument, you will see that not a single instance is documented where both premises are true and the conclusion is false. (You will also notice that not a single a instance is documented where both premises are true, hence in this case, all arguments with contradictory premises are valid by definition as it is impossible for such arguments to contain all true premises and a false conclusion. An argument is invalid if and only if it contains a single instance of all true premises and a false conclusion.)

Premise one: A
Premise two: Not A.

Conclusion: B

In the argument with contradictory premises, it is possible to have one or more false premises that lead to a true conclusion.

-----------------------------------------------

In short, I endorse reason's view that it is possible to have an argument with false premises that lead to a contradiction, yet I think the argument he cited does not adequately exemplify this concept. An argument with contradictory premises is the only example of this phenomen that I can think of. Such an argument is deductively valid, but always epistemically unacceptable as such a reasoning process is ostensibly preposterous whether it yields the true conclusion or not. This is so because a reasoning process of this kind can entail any conclusion.

Originally Posted by Evan

So it can go from general to general, specific to specific, or general to specific, but never specific to general (that's induction).

Although you are onto something with the supposition that induction is typically more general than deduction and in most cases deductive reasoning goes from general to the specific, your claim that a deductive reasoning process never leads from specific to general is false.

A general proposition is one that contain many components and is typically opaque in meaning. A specific proposition is one that contains very few components and tends to be specific in meaning.

Pay careful attention to this reasoning process.

Premise 1: A
Premise 2: If A then B (A horseshoe B in the language of Symbolic Logic)
Step 1: B (1,2 Modus Ponens)
Conclusion: B or A ( Or is used in an inclusive sense in symbolic logic. Such a statement is merely saying that at least one of the two terms is true. Could be just one, it also could be both. Hence, when I prove that B is true, I can infer the proposition of B or Anything. I could have as justifiably stated B or not A, or B or Z)

Another example.

Premise 1: If A and B.
Step 1: A
Step 2: B
Conclusion: if A then B. (Once you have proved B, you can state that if anything then B, as in material implication the antecedent serves as just one possible reason for the verity of the consequence. If the consequence is true, the truthfulness of the antecedent is no longer relevant as the consequent may be true for reasons other than the verity of the antecedent. This is not to be confused with the triple bar symbol, or logical equivalence. In that case the antecedent serves as the only reason for the truthfulness or falsity of the conclusion. In the expression of A if and only if B (triple Bar), B is true only if A is and A is true only if B is. )

In both examples we start with very specific premises (straightforward and simple) and arrive at a conclusion that is very general. Because the examples above evince that it is possible for deduction to proceed from specific to general, we know that it is possible for deduction to proceed from specific to general and vice versa. (Its a truism that deduction can go from general to specific.)

"Do not argue with an idiot. They drag you down to their level and beat you with experience." -- Mark Twain

Deduction can't be adequately defined as a mode of inference from the general to the specific, because only in some cases does this characterisation hold.

The matter is one of logical strength or content. In a deduction, the premises must have greater logical strength than the conclusion, i.e. the conclusion cannot entail anything which the premises cannot also entail. Or in other words, the logical content of the conclusion must be a subset of the premises. In some deductive arguments (quantitive relations), this can manifest itself as inference from the general to specific, but it need not for all cases of deduction.

NOTE: Regarding my former comments, this is why the relation between induction and deduction is not like addition and substraction. For example, consider trying to induce the premises of the first argument above from its conclusion. I don't think even the most ardent inductivist considers that an "inductively valid" inference, and yet its reverse is valid deductively.

A criticism that can be brought against everything ought not to be brought against anything.

First, it is possible to validly infer a universal statement from a set of premises including only singular statements. For example,

x is y |= All x are y or not-y

This only works when the conclusion is a tautology (and, therefore, even follows from the empty set of premises), but should be noted as a caveat to previous comments.

Ah. Sure. But then you could rewrite the premises to be general too. I guess that means my previous statement is kinda useless. I probably should have specified non-tautologies.

Originally Posted by SolitaryWalker

"Every hamster is a member of TypologyCentral
Provoker is a hamster
Therefore,
Provoker is a member of TypologyCentral"

This argument is guilty of the informal logical fallacy of equivocation. Logic has two components, extensions and intensions. Extensions are concerned with how entities of discourse relate to each other. (In accordance to laws of reasoning, for instance, demorgan's law, modus ponens, modus tollens and so on.) Intensions are concerned with how the terms of discourse are defined.

When I say that provoker is a member of typologycentral, I define provoker as a person who made several posts on this forum. If I say that provoker is a hamster, I mean something entirely different. Hence, in the argument above, the conclusion is false. A hamster named provoker is not a member of typologycentral.

Huh?

You seem to think his conclusion is "provoker is a hamster and a member of typecentral" instead of just what it says.